The present invention relates to methods for determining transport properties of fluids in porous media by using nuclear magnetic resonance (NMR) in combination with pulsed magnetic field gradients (PFG). Examples of transport properties include but are not limited to the measurement of the self diffusion coefficients and mean square displacements of the fluid molecules, the relative amounts of fluid molecules having a particular self diffusion coefficient and mean square displacement, respectively and the pore size distribution of the porous medium. In particular the present invention relates to measuring the bound and free fluid volume fractions, also known as the index, of fluids in rocks by measuring the self diffusion coefficients and mean square displacements of both fluid fractions as a function of the observation time and/or the fluid saturation in the pulsed field gradient nuclear magnetic resonance (PFG NMR) experiment.
The free fluid index (FFI) refers to the fluid fraction in a rock pore space, which can be produced under typical reservoir production conditions. The FFI is also sometimes refered to as the "movable" fluid fraction. The bound fluid index (BFI) is the fluid fraction which can not be recovered from the reservoir rock. The petrophysical standard method to determine the FFI and BFI in laboratory measurements on reservoir core plugs is the determination of the capillary pressure curve of the fluid in the pore space by progressive desaturation of the core plug using successively higher rotation speeds of the core plug in a centrifuge. The BFI is the fluid fraction remaining in the pore space of the rock core at a certain capillary pressure i.e. at a certain rotation speed of the centrifuge. The FFI is the fluid fraction which has been removed from the rock core at this rotation speed. Alternatively, the capillary pressure curve and therefore the BFI and FFI can be obtained from mercury injection experiments which also measures the pore size distribution of the rock.
Another approach for determining the BFI and FFI in rocks is the measurement of the transverse (T.sub.2) or longitudinal (T.sub.1) nuclear magnetic relaxation times of fluids in rocks [see C. Straley et. al. "NMR in Partially Saturated Rocks: Laboratory Insights on Free Fluid Index and Comparison with Borehole Logs", SPWLA 32nd Annual Logging Symposium, Jun. 16-19, 1991]. It has been shown that under the condition of the fast diffusion limit, the nuclear magnetic relaxation times of fluids in a single pore depend only on the surface to volume ratio of that pore and is therefore a measure of the pore size [see M. H. Cohen and K. S. Mendelson, "Nuclear Magnetic Relaxation and the Internal Geometry of Sedimentary Rocks", J. Appl. Phys. vol. 53, pg. 1127, (1982)]. Since rocks are typically described by a wide range of pore sizes, the observed magnetization decay in the NMR experiment is muli-exponential. The distribution of relaxation times describing this decay is a measure of the pore size distribution in the rock. In this approach, the BFI is the fraction of the fluid having relaxation times less than a certain T.sub.2 or T.sub.1 cut-off value. These actual cut-off values depend on the effectiveness of the surface relaxation, which is in general, expected to be a function of the detailed properties such as the mineralogy and surface roughness of the rock.
Nuclear magnetic resonance has been used for some time to study fluid flow and diffusion [P. T. Callaghan, Principles of Nuclear Magnetic Resonance, Clarendon Press, Oxford 1991]. In general, molecular displacements of fluid molecules can be quantified with NMR by using pulsed magnetic field gradients. With the application of a magnetic field gradient G, the precession frequency .omega. of a nuclear magnetic moment in an external magnetic field B=(0, 0, B.sub.o) is a function of the projection of the position r of the magnetic moment on the direction of the applied field gradient: EQU .omega.=.gamma.(B.sub.o +G.multidot.r) (1)
In the PFG NMR experiment the amplitude of the NMR signal, such as a spin echo, is measured as a function of the intensity of applied pulsed magnetic field gradients. Pulse timing diagrams for the applied radio frequency pulses and magnetic field pulses are shown in FIG. 1. The position of each fluid molecule is encoded by the first magnetic field gradient pulse of duration .delta. and strength g. After an elapsed time .DELTA., a second gradient pulse is applied during the refocussing period. The latter is defined by the time period following the second radio frequency pulse in the timing diagram in FIG. 1. The detected signal is the NMR spin echo which is formed at the end of the refocusing period. For stationary spins the phase acquired during the first gradient pulse is reversed by the second pulse which should be matched in intensity g.delta.. For moving spins the phase reversal is incomplete depending on the distance the molecule has moved during the time between the two gradient pulses. A displacement causes a change in the NMR resonance frequency, or equivalently a change in the phase of the signal, which leads to an attenuation of the observed spin echo amplitude. By repeating the experiment and systematically incrementing the intensity of the matched field gradient pulses one obtains a set of spin echo amplitudes M(g.delta.,.DELTA.) with an attenuation .psi.(g.delta.,.DELTA.) which is characteristic for the displacement of the spins. The spin echo attenuation in the PFG NMR is given by [P. T. Callaghan, Principles of Nuclear Magnetic Resonance, Clarendon Press, Oxford 1991]: ##EQU1## where P(r.vertline.r', .DELTA.)dr' denotes the conditional probability (propagator) of finding a molecule initially at position r, after a time .DELTA. in the volume element dr' at r' and p(r) is the initial spin density. In a homogeneous medium, the propagator P(r.vertline.r',.DELTA.) is a Gausian function of the displacement (r'--r). Therefore, the spin echo attenuation becomes a exponential function of the square of the intensity of the applied field gradient pulses with the self diffusion coefficient D as decay rate: EQU .psi.(g.delta.,.DELTA.)=exp[-(.gamma..delta.g).sup.2 D.DELTA.](3)
For self-diffusion in restricted geometries it has been shown that deviations from the pattern predicted by eq. [3] occur at high intensities of the applied field gradients where the spin echo amplitude has already been attenuated by 2-3 orders of magnitude [P. T. Callaghan et. al., Nature, 351,467 (1991)]. With increasing observation time these effects become more pronounced since more molecules will experience restrictions of their translational motion due to the pore walls. The pore geometry and a measure of the pore sizes can be estimated from this non-exponential spin echo attenuation [see for example, P. S. Sen and M. D. Hurlimann, J. Chem Phys., 101, 5423 (1994). However, at small (.gamma..delta.g).sup.2 eq. [3] may still be applied yielding a observation time dependent apparent self diffusion coefficient D, which contains information on pore sizes of the porous medium.
The present invention provides a method for obtaining at least one transport property of fluids in porous media by encoding the self diffusion of the fluid molecules in the pore space by PFG NMR.
In one embodiment of the present invention, the time dependence of the mean square displacements of the fluid molecules are obtained, which show that a fraction of the molecules experiences a highly restricted self diffusion (low translational mobility) during the time scale of the PFG NMR experiment while the other fraction shows a high translational mobility with only slight deviations from the self diffusion coefficient of the bulk fluid. The ratio of the fluid with low translational mobility to the overall fluid is shown to be the BFI. The pore size of the rock as seen by both fluid fractions may be estimated from the time dependence of the mean square displacements.